McMullen [1]} used Lemma REF  to give a lower bound
for the dimension of a set \(E\)  constructed this way in terms of
\(\sup _{V\in \mathcal {E}_m}\operatorname{diam}V\)  and \(\inf _{V\in \mathcal {E}_m}\operatorname{dens}(E_{m+1},V)\) .
Adapting some ideas from [2]},
we will instead work with bounds for \(\operatorname{diam}V\)  and \(\operatorname{dens}(E_{m+1},V)\) 
which depend not only on \(m\) , but also on \(V\) .
